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Wave breaking in the surf zone and deep water in a
non-hydrostatic model
Morteza Derakhti1, James T. Kirby1, Fengyan Shi1 and Gangfeng Ma2
1Center for Applied Coastal Research, University of Delaware, Newark, DE, USA2Department of Civil and Environmental Engineering, Old Dominion University,
Norfolk, VA, USACorresponding author: Email address: [email protected]
Abstract
We examine wave-breaking predictions ranging from shallow to deep water
conditions using a non-hydrostatic model NHWAVE (Ma et al., 2012; Der-
akhti et al., 2015), comparing results both with corresponding experiments
and with the results of a volume-of-fluid (VOF)/Navier-Stokes solver (Ma
et al., 2011; Derakhti & Kirby, 2014a,b). Our study includes regular and ir-
regular depth-limited breaking waves on planar and barred beaches as well as
steepness-limited unsteady breaking waves in intermediate and deep water.
Results show that the model accurately resolves breaking wave properties in
terms of (1) time-dependent free-surface and velocity field evolution, (2) inte-
gral breaking-induced dissipation, (3) second- and third-order wave statistics,
(4) time-averaged breaking-induced velocity field, and (5) turbulence statis-
tics in depth-limited breaking waves both on planar and barred beaches.
The breaking-induced dissipation is mainly captured by the k− ε turbulence
model and involves no ad-hoc treatment, such as imposing hydrostatic con-
Preprint submitted to Ocean Modelling June 26, 2015
ditions. In steepness-limited unsteady breaking waves, the turbulence model
has not been triggered, and all the dissipation is imposed indirectly by the
TVD shock-capturing scheme. Although the absence of turbulence in the
steepness-limited unsteady breaking events which leads to the underestima-
tion of the total breaking-induced dissipation, and, thus, the overprediction
of the velocity and vorticity field in the breaking region, the model is capable
of predicting (1) the dispersive and nonlinear properties of different wave
packet components before and after the break point, (2) the overall wave
height decay and spectral evolution, and (3) the structure of the mean ve-
locity and vorticity fields including large breaking-induced coherent vortices.
The same equations and numerical methods are used for the various depth
regimes, and vertical grid resolution in all simulated cases is at least an order
of magnitude coarser than that of typical VOF-based simulations.
Keywords:
non-hydrostatic wave model, breaking waves, shock-capturing
1. Introduction1
One of the least understood and yet most important events in the ocean2
upper layer is the breaking of surface waves. Surface wave breaking, a3
complex, two-phase flow phenomenon, plays an important role in numer-4
ous environmental and technical processes such as air-sea interaction, acous-5
tic underwater communications, optical properties of the water columns,6
nearshore mixing and coastal morphodynamics. Surface wave breaking is one7
2
of the most challenging process in coastal hydrodynamic modeling. Model8
results become even more dubious and problematic as model resolution de-9
creases. During active breaking, perhaps the major simplification by any10
non-hydrostatic model is achieved by replacing a complex free surface by a11
single-valued function of horizontal location. Instead of having a jet/splash12
cycle in plunging breakers or formation of surface rollers and a turbulent13
bore in spilling breakers, this simplification leads to the formation of a rela-14
tively sharp wave-front, analogous to a jump discontinuity in a shock-front15
propagation, as a wave approaches breaking. The sharp wave-front prop-16
agates without any unphysical numerical oscillation when an appropriate17
shock-capturing scheme is used.18
Although turbulence-resolving frameworks such as large-eddy simulations19
(LES) combined with the volume-of-fluid (VOF) method for free-surface20
tracking (Watanabe et al., 2005; Lakehal & Liovic, 2011; Derakhti & Kirby,21
2014a; Zhou et al., 2014; Lubin & Glockner, 2015) can resolve small scale22
processes such as breaking-induced turbulent coherent structures, they are23
still computationally expensive even for laboratory-scale events. A lower-24
resolution three-dimensional (3D) framework is needed to study long-term,25
O(hrs), and large-scale, O(100m ≈ 10km), breaking-driven circulation as26
well as transport of sediment, bubbles, and other suspended materials. Dur-27
ing the past decade, several 3D wave-resolving non-hydrostatic models based28
on Reynolds-averaged Navier-Stokes (RANS) equations have been developed29
for coastal applications (Ma et al., 2012; Young & Wu, 2010; Zijlema et al.,30
3
2011; Bradford, 2011; Shirkavand & Badiei, 2014).31
For surf zone breaking waves, when non-hydrostatic effects are retained,32
Smit et al. (2013) have emphasized that high resolution in the vertical di-33
rection (more than 15 levels) is needed for reasonable integral dissipation34
and corresponding wave-height decay resulting from discontinuity propaga-35
tion. In place of common shock-capturing schemes (Toro, 2009), they used a36
special treatment to maintain momentum conservation across flow disconti-37
nuity, observing that insufficient vertical resolution led to an underestimation38
of velocities, thereby delaying the initiation of breaking. They proposed a hy-39
drostatic front approximation in which the non-hydrostatic part of pressure is40
switched off by analogy to the nonlinear shallow water equations. Using this41
technique, SWASH was shown to predict the evolution of wave-height statis-42
tics in a surf zone reasonably well compared with laboratory measurements43
of irregular waves on a plane slope, by using a few σ levels. In the present44
study, however, we will show that NHWAVE, as described in Derakhti et al.45
(2015), accurately captures the wave-height decay in regular waves as well as46
wave-height statistics in irregular surf zone breaking waves using as few as 447
vertical σ levels, without recourse to disabling of non-hydrostatic effects.48
Organized flow structures and their evolution have a critical role in long-49
term mixing and transport of fine sediment, bubbles, and other suspended50
materials in the ocean upper layer and surf zone. For example, large co-51
herent vortices induced by individual whitecaps in deep and intermediate52
water (Rapp & Melville, 1990; Pizzo & Melville, 2013; Derakhti & Kirby,53
4
2014b) as well as undertow, longshore and rip currents (Longuet-Higgins,54
1970; Svendsen, 1984) in the surf zone are fairly well-understood breaking-55
induced organized motions. Such organized motions need to be reasonably56
resolved in any RANS-based framework to truly estimate long-term trans-57
port and mixing processes at field scales. The effect of Langmuir circulation58
cells should also be taken into account in deep water mixing. The available59
relevant literature on non-hydrostatic models mainly are related to surf zone60
breaking waves (or depth-limited breaking waves) and mostly focus on the61
capability of these models to predict free surface evolution and wave statis-62
tics, while less attention has been dedicated to velocity and turbulence fields.63
Although there are recent studies (Young & Wu, 2010; Ai et al., 2014) exam-64
ining the capability of non-hydrostatic models to resolve wave-wave nonlinear65
interaction and dispersion properties of non-breaking deep water waves, no66
study has examined non-hydrostatic model predictions of breaking-related67
processes in steepness-limited unsteady breaking waves.68
Our goals here are (1) to carefully examine what level of detail of a veloc-69
ity field and of turbulence statistics can be reproduced by the non-hydrostatic70
model NHWAVE as described by Derakhti et al. (2015), across the inner shelf71
and nearshore regions, and (2) to establish whether this models is capable72
of providing accurate representations of breaking-wave properties in inter-73
mediate/deep water. Model results for regular and irregular depth-limited74
breaking waves over planar and barred beaches as well as steepness-limited75
unsteady breaking waves generated by the dispersive focusing technique will76
5
be presented in detail, focusing on wave-breaking-related large-scale processes77
categorized as (1) time dependent free-surface and mean velocity field evo-78
lution, (2) integral breaking-induced dissipation, (3) second- and third-order79
wave statistics, (4) wave-averaged breaking-induced organized velocity field,80
and (5) ensemble-averaged breaking-induced turbulence statistics.81
The paper is organized as follows. A brief description of the model is pre-82
sented in §2. Details of the numerical set-up, and comparisons of model re-83
sults with measurements for depth-limited breaking waves on a planar beach84
and on a barred beach are given in §3 and §4 respectively. The numerical set-85
ups and comparisons of model results with measurements and with results of86
LES/VOF simulations of Derakhti & Kirby (2014a,b) for steepness-limited87
unsteady breaking waves are given in §5. Discussions and conclusions are88
presented in §6.89
2. Mathematical formulation and numerical methods90
The non-hydrostatic model NHWAVE is originally described in Ma et al.91
(2012). NHWAVE solves the RANS equations in well-balanced conserva-92
tive form, formulated in time-dependent, surface and terrain-following σ93
coordinates. The governing equations are discretized by a combined finite-94
volume/finite-difference approach with a Godunov-type shock-capturing scheme.95
The model is wave-resolving and can provide instantaneous descriptions of96
surface displacement and wave orbital velocities. The model has been ap-97
plied to study tsunami wave generation by submarine landslides (Ma et al.,98
6
2013a; Tappin et al., 2014), wave damping in vegetated environments (Ma99
et al., 2013b), nearshore suspended sediment transport (Ma et al., 2014a),100
and wave interaction with porous structures (Ma et al., 2014b). In these101
studies, the effects of surface and bottom slopes in the dynamic boundary102
conditions (Ma et al., 2012, §3), as well as in the horizontal diffusion terms103
of the transport equation for suspended sediment concentration (Ma et al.,104
2013a, equation 10) and k − ε equations (Ma et al., 2013a, equations 13,14)105
were ignored. Derakhti et al. (2015) have recently derived a new form of the106
governing equations together with the exact surface and bottom boundary107
conditions. They have shown that surface slope effects should be taken into108
account in order to accurately resolve turbulence statistics, such as turbulent109
kinetic energy (k) distribution, in surf zone breaking waves. Here, we use the110
Derakhti et al. (2015) formulation together with the k−ε model based on the111
renormalization group theory (Yakhot et al., 1992). The reader is referred to112
Derakhti et al. (2015) for the details of the governing equations, surface and113
bottom boundary conditions and numerical methods.114
3. Depth-limited breaking waves on a planar beach115
In this section, we consider model performance for the case of regular116
and irregular depth-limited wave breaking on a planar beach using the data117
sets of Ting & Kirby (1994) for regular waves and of Bowen & Kirby (1994)118
and Mase & Kirby (1992) for irregular waves. All experiments have been119
conducted in wave flumes approximately 40m long, 0.6m wide and 1.0m deep.120
7
Results for regular and irregular wave breaking cases are given in §3.1 and121
§3.2, respectively. In each section, the experimental and numerical set-ups122
for the corresponding cases will be described.123
3.1. Regular breaking waves124
Both spilling breaking (hereafter referred as TK1) and plunging breaking125
(hereafter referred as TK2) cases of Ting & Kirby (1994) are selected to126
examine the model capability and accuracy to reproduce the free surface and127
mean velocity field evolution, breaking-induced wave-averaged velocity field128
and k estimates. This experiment has been widely used by other researchers129
to validate both non-hydrostatic (Ma et al., 2014a; Bradford, 2011, 2012;130
Smit et al., 2013; Shirkavand & Badiei, 2014) and VOF-based (Ma et al.,131
2011; Lin & Liu, 1998; Bradford, 2000; Christensen, 2006; Lakehal & Liovic,132
2011) numerical models. Figure 1 sketches the experimental layout and the133
cross-shore locations of the available velocity measurements. The velocity134
measurements were obtained using Laser Doppler velocimetry (LDV) along135
the centerline of the wave tank. Table 1 summarizes the input parameters136
for TK1 and TK2.137
A uniform grid of ∆x = 0.025m is used in the horizontal direction. Grids138
with 4, 8, and 16 uniformly spaced σ levels are used to examine the effects of139
varying vertical resolution. At the inflow boundary, the free surface location140
and velocities are calculated using the theoretical relations for cnoidal waves141
as given in Wiegel (1960). The right end of the numerical domain is extended142
8
Table 1: Input parameters for the simulated surf zone regular breaking cases on a planarbeach. Here, d0 is the still water depth in the constant-depth region, H and T are thewave height and period of the cnoidal wave generated by the wavemaker, (kH)0 is thecorresponding deep water wave steepness of the generated wave, ξ0 = s/
√H0/L0 is the
self similarity parameter, and s is the plane slope.
Case no. d0 H T (kH)0 ξ0 breaking(m) (m) (s) type
TK1 0.4 0.125 2.0 0.126 0.20 spillingTK2 0.4 0.128 5.0 0.015 0.59 plunging
beyond the maximum run-up, and the wetting/drying cells are treated as de-143
scribed in Ma et al. (2012, §3.4) by setting Dmin = 0.001m. In this section, 〈 〉144
and ( ) refer to phase and time averaging over five subsequent waves after the145
results reach quasi-steady state, respectively. The corresponding measured146
averaged variables, were calculated by averaging over 102 successive waves147
starting at a minimum of 20 minutes after the initial wavemaker movement.148
The mean depth is defined as h = d+ η, where d is the still water depth149
and η is the wave set-down/set-up. Here, x = 0 is the cross-shore location150
at which d = 0.38m as in Ting & Kirby (1994), and x∗ = x − xb is the151
horizontal distance from the initial break point, xb. In Ting & Kirby (1994),152
the break point for spilling breakers was defined as the location where air153
bubbles begin to be entrained in the wave crest (xb = 6.40m), whereas for154
plunging breakers it was defined as the point where the front face of the wave155
becomes nearly vertical (xb = 7.795m). In the model the break point is taken156
to be the cross-shore location at which the wave height starts to decrease,157
approximately 0.7m seaward of the observed xb for both TK1 and TK2.158
9
3.1.1. Time-dependent free surface evolution159
Figure 2 shows the cross-shore distribution of crest, 〈η〉max, and trough,160
〈η〉min, elevations as well as mean water level, η in the shoaling, transition161
and inner surf zone regions for the spilling case TK1 and plunging case TK2.162
Figures 3 and 4 show the phase-averaged water surface elevations at different163
cross-shore locations before and after the initial break point for TK1 and164
TK2, respectively. In the shoaling and inner surf zone regions, the model165
captures the water surface evolution reasonably well in both cases. The166
predicted cross-shore location of the initial break point, however, is slightly167
seaward of the measured location for both cases, regardless of the choice of168
vertical resolution (Figure 2 a,b), as in the two-dimensional (2D) VOF-based169
simulations (Bradford, 2000, Figures 1 and 7). In both cases, after shifting170
the results with respect to the cross-shore location of the break point, the171
model captured the free surface evolution, wave height decay rate (Figure172
2A,B), crest and trough elevations, as well as wave set-up reasonably well173
using as few as 4 σ levels.174
3.1.2. Organized flow field175
Figures 5 and 6 show the oscillatory part of the phase-averaged horizontal176
velocities 〈u〉− u normalized by the local phase speed√gh, at different cross-177
shore locations in the shoaling, transition and inner surf zone regions at about178
5cm above the bed for TK1 and TK2, respectively. In general, the model179
captures the evolution of 〈u〉 − u fairly reasonably both in time and space180
10
in both cases using as few as 4 σ levels, and the predicted 〈u〉 − u of the181
simulations with different vertical resolutions are nearly the same. For the182
spilling case (Figure 5) there is an apparent landward increasing phase lead183
in the results of the simulation with 4 σ levels, indicating an overestimation184
of bore propagation speed at low vertical resolutions. This error is corrected185
at the higher resolutions of 8 and 16 σ levels.186
Figure 7 shows the spatial distribution of the time-averaged velocity field187
using different vertical resolutions for TK1. To obtain the Eulerian mean ve-188
locities, the model results in the σ-coordinate system first were interpolated189
onto a fixed vertical mesh at each cross-shore location using linear interpola-190
tion, and then time averaging was performed. The predicted return current191
using 4 σ levels shown in 7(a) has not detached from the bed at x∗ ∼ 0192
in contrast to the simulations with 8 and 16 σ levels. The results of the193
simulations with different vertical resolutions have approximately the same194
structure in the surf zone. A similar pattern of results was found for the195
plunging case TK2 and is not shown.196
The amount of curvature in the predicted undertow profiles is greater197
than in the measured undertow profiles for both cases, as shown in Figures198
8 and 9. This difference is more noticeable in the plunging case TK2, in199
which the measured profiles are approximately uniform with depth. Con-200
sidering available undertow models using an eddy viscosity closure scheme201
(see Garcez Faria et al., 2000, among others), it is known that the three fac-202
tors determine the vertical profile of undertow currents; including (i) bottom203
11
boundary layer (BBL) processes, leading to a landward streaming velocity204
(Longuet-Higgins, 1953; Phillips, 1977) or a seaward streaming velocity due205
to a time-varying eddy viscosity within the wave turbulent BBL (Trowbridge206
& Madsen, 1984), close to the bed; (ii) vertical variations of the eddy viscosity207
νt, affected mainly by breaking-generated turbulence; and (iii) wave forcing208
due to the cross-shore gradients of radiation stress, set-up, and convective209
acceleration of the depth-averaged undertow. As explained by Garcez Faria210
et al. (2000), the amount of curvature in the undertow profile is a function211
of both wave forcing and νt. Large values of wave forcing generates more212
vertical shear, resulting in a parabolic profile, whereas large values of νt re-213
duce vertical shear, leading to a more uniform velocity profile with depth.214
As shown in the next section, we believe that the underprediction of turbu-215
lence, and, thus, the underprediction of νt results in greater vertical shear in216
the predicted undertow profiles, where the larger discrepancy in TK2 is due217
to the more noticeable underprediction of νt in TK2 compared with that in218
TK1. In addition, the difference between the predicted and measured return219
velocities close to the bed have relatively larger deviations in TK2 than in220
TK1. This may be due to the lack of second-order BBL effects, and, thus,221
the absence of the associated streaming velocity, in the present simulations.222
Compared with measurements, the model predicts the time-averaged Eu-223
lerian horizontal velocity field fairly reasonably using as few as 4 σ levels for224
both cases.225
12
3.1.3. Turbulence Statistics226
Figure 10 shows snapshots of the predicted instantaneous k distribution227
using 4 and 8 σ levels for TK1. Increasing the vertical resolution decreases228
the predicted k levels in the transition region and increases k in the inner229
surf zone. Generally, the overall distribution of k is the same. The same230
trend is also observed for TK2 (not shown).231
Figure 11 shows a comparison of modeled and measured 〈k〉 time series232
at about 4cm and 9cm above the bed at different cross-shore locations using233
4, 8 and 16 σ levels for TK1. Comparing different resolutions, a reasonable234
〈k〉 level at different cross-shore locations is captured by the model using as235
few as 4 σ levels. 〈k〉 is overestimated higher in the water column during the236
entire wave period especially close to the break point. This overestimation237
has been also reported in previous VOF-based k−ε studies (Lin & Liu, 1998;238
Ma et al., 2011). Lin & Liu (1998) argued that this is because the RANS239
simulation can not accurately predict the initiation of turbulence in a rapidly240
distorted shear flow such as breaking waves. Alternately, Ma et al. (2011)241
incorporated bubble effects into the conventional single phase k − ε model,242
and concluded that the exclusion of bubble-induced turbulence suppression243
is the main reason for the overestimation of turbulence intensity by single244
phase k − ε. Comparing Figure 11 with the corresponding results from the245
VOF-based model Ma et al. (2011, Figure 7), we can conclude that predicted246
〈k〉 values under spilling breaking waves by NHWAVE are at least as accurate247
as the VOF-based simulation without bubbles.248
13
In the plunging case TK2, a different behavior is observed in the predicted249
〈k〉 values shown in Figure 12 compared with the corresponding results for250
TK1, regardless of the various vertical resolutions. After the initial break251
point, 〈k〉 is underpredicted especially for lower elevations. Figure 12 shows252
〈k〉 time series at 4cm and 9cm above the bed as well as the corresponding253
measurements of Ting & Kirby (1994) for TK2. The model could not resolve254
the sudden injection of k into the deeper depths at the initial stage of active255
breaking, and, thus, there is a considerable underprediction of 〈k〉 at the256
beginning of active breaking below trough level.257
Figure 13 shows k field using 4, 8 and 16 σ levels for TK1. The increase258
of the vertical resolution leads to a more concentrated patch of k. A similar259
trend is also observed for TK2 (not shown). Figures 14 and 15 show the260
comparison of modeled and measured k profiles at different cross-shore loca-261
tions before and after the initial break point for TK1 and TK2 respectively.262
For TK2, the noticeable underprediction of 〈k〉 at the initial stage of active263
breaking shown in Figure 12 compensates relatively smaller overprediction264
of 〈k〉 at the other phases, resulting to apparent smaller k values than those265
in the measurement in the shoreward end of the transition region and inner266
surf zone, as shown in Figure 15(d-g).267
It can be concluded that the vertical resolution of 4 σ levels is sufficient268
to capture the temporal and spatial evolutions of k for the spilling case269
TK1. For the plunging case TK2, the vertical advection of k into the deeper270
depths can not be captured by increasing the σ levels, and, thus, k is always271
14
underpredicted at those depths.272
3.2. Irregular breaking waves273
In this section, we use one of three cases of Bowen & Kirby (1994) (here-274
after referred as BK) and both cases of Mase & Kirby (1992) (hereafter re-275
ferred as MK1 and MK2) in order to compare the model predictions of power276
spectra evolution, integral breaking-induced dissipation and wave statistics277
of the surf zone breaking irregular waves on a planar beach. The three cases278
have different dispersive and nonlinear characteristics as summarized in Table279
2. The data set of Mase & Kirby (1992) has been used in a number of pre-280
vious studies of spectral wave modeling in the surf zone. In particular, MK2281
has a high relative depth of kpd0 ∼ 2 at the constant-depth region and a high282
relative steepness of (kpHrms)0 ∼ 0.16, and thus, is a highly dispersive and283
nonlinear case. In these two experiments, irregular waves with single-peaked284
spectra were generated and allowed to propagate over a sloping planar bot-285
tom. Figures 16 and 17 sketch the corresponding experimental layouts and286
the cross-shore locations of the available free surface measurements. Bowen287
& Kirby (1994) used a TMA spectrum with a width parameter γ = 3.3 to288
generate the initial condition at the wavemaker. In Mase & Kirby (1992),289
random waves were simulated using the Pierson-Moskowitz spectrum.290
Uniform grid of ∆x = 0.025m, 0.015m and 0.01m is used in the horizon-291
tal direction for BK, MK1 and MK2 cases, respectively. Resolutions of 4292
and 8 σ levels are used to examine the effects of different vertical resolution.293
15
Table 2: Input parameters for the simulated surf zone irregular breaking cases on a planarbeach. Here, d0 is the still water depth in the constant-depth region, kpd0 and (kpHrms)0are the dispersion and nonlinearity measure of the incident irregular waves respectively,fp is the peak frequency of the input signal, ξ0 = s/
√(Hrms)0/L0 is the self similarity
parameter, L0 = g(2π)−1f−2p , and s is the plane slope.
Case no. d0 kpd0 (kpHrms)0 fp ξ0 dominated(m) (Hz) breaking type
BK 0.44 0.30 0.016 0.225 0.56 plungingMK1 0.47 0.93 0.058 0.6 0.52 plungingMK2 0.47 1.97 0.161 1.0 0.31 spilling
The cross-shore location of the numerical wavemaker is set to be the first294
gage location. The measured free surface and velocities determined from295
linear theory are constructed at the wavemaker using the first 5000 Fourier296
components of the measured free surface time series. The right end of the297
numerical domain is extended beyond the maximum run-up, and the wet-298
ting/drying cells are treated as described in Ma et al. (2012, §3.4) by setting299
Dmin = 0.001m. In this section, ( ) refers to long-time averaging over several300
minutes, more than 300 waves. The first 1000 data points were ignored both301
in the model result and the corresponding experiment for all cases. The mean302
see level is defined as h = d+ η, where d is the still water depth and η is the303
wave set-down/set-up. Here, x∗ = x− xb is the horizontal distance from the304
xb, we define as the cross-shore location in which Hrms is maximum.305
3.2.1. Power spectra evolution and integral breaking-induced dissipation306
The shape and energy content of wave spectra in nearshore regions are307
observed to have a considerable spatial variation over distances on the order308
16
of a few wavelengths due to continued wave breaking-induced dissipation as309
well as triad nonlinear interactions between different spectral components310
(Elgar & Guza, 1985; Mase & Kirby, 1992). Here, we will examine the311
model prediction of the integral breaking-induced dissipation compared with312
the corresponding measurements by looking at the evolution of the power313
spectral density, S(f), from outside the surf zone up to the swash region.314
Figure 18 shows the variation of the computed S(f) using 4 and 8 σ315
levels for the random breaking cases, BK, MK1 and MK2, as well as the316
corresponding measured S(f). The measured signals were split into 2048317
data points segments. Each segment multiplied by a cosine-taper window318
with the taper ratio of 0.05 to reduce the end effects. The measured spectrum319
is obtained by ensemble averaging over the computed spectra of 11, 8, 7320
segments for BK, MK1 and MK2 respectively and then band averaging over321
5 neighboring bands. The resultant averaged spectra of BK, MK1 and MK2322
have 110, 80 and 70 degrees of freedom, respectively. The sampling rate was323
25 Hz (fNyq = 12.5Hz) for BK and MK1 and 20 Hz (fNyq = 10Hz) for MK2.324
The spectral resolution for BK, MK1 and MK2 are ∆f = 0.06Hz, 0.06Hz and325
0.05Hz, respectively. The spectrum for the computed wave field is obtained326
in a similar way, with the same spectral resolution and degrees of freedom.327
The first two rows of Figure 18 show S(f) outside the surf zone, while the328
other panels cover the entire surf zone up to a shallowest depth of d ∼ 3cm.329
Comparing with the measurements, the model captures the evolution of S(f)330
in the shoaling region as well as in the surf zone fairly well. We used the331
17
measured surface elevation time series at d = d0 as an input, and, thus,332
the infra-gravity waves are introduced in the domain as in the experiment.333
The more pronounced predicted energy at this frequency range (f/fp ≈ 0.5)334
compared with measurements at shoreward cross-shore locations is due to335
the absence of lateral side walls effects and the reflection from the upstream336
numerical boundary, which is located closer than the physical wavemaker337
used in the experiment to the plane slope, especially in MK1 and MK2. In338
addition the input low frequency climate is not exactly the same as in the339
measurement. The reason is that, we impose the input low frequency signal340
as a progressive wave at the numerical boundary while it was a standing wave341
in the measurement.342
We can conclude that the integral breaking-induced dissipation is cap-343
tured by the model, using as few as 4 σ levels. In addition, an asymptotic344
f−2 spectral shape of the wave spectrum in the inner surf zone (Kaihatu345
et al., 2007), due to the sawtooth-like shape of surf zone waves, is fairly346
reasonably captured by the model in all cases.347
3.2.2. Wave statistics348
Second-order wave statistics such as a significant wave height and a sig-349
nificant wave period, characterize the relative strength/forcing of irregular350
waves which need to be estimated for different coastal/inner-shelf related351
calculations and designs. These may be defined based on the wave spectrum,352
S(f), as a significant wave height Hm0 = 4m1/20 and the mean zero-crossing353
18
period Tm02 = (m0/m2)1/2, where mn =
∫fnS(f)df , is the nth order mo-354
ment of S(f), or based on the statistics of a fairly large number of waves355
(Figure 19, first row) extracted from the associated surface elevation time356
series by using the zero-up crossing method. The second and third rows of357
Figure 19 show the cross-shore variations of the model predictions of η, Hm0 ,358
Tm02 together with H1/10 and T1/10 which represent the averaged wave height359
and period of the one-tenth highest waves, using 4 and 8 σ levels as well as360
the corresponding measured values for the random breaking cases, BK, MK1361
and MK2. At the very shallow depths d < 0.05cm the model predictions of362
H1/10 and T1/10 deviates considerably from the measurements. This devia-363
tion is mainly due to the relatively higher energy of infra-gravity waves in364
the model results compared with that in the measurements, as discussed in365
the previous section. To eliminate the infra-gravity and very high frequency366
wave effects, both the measured and computed ensemble-averaged S(f) have367
been band-pass filtered with limits 0.25fp < f < 8.0fp, and then Hm0 and368
Tm02 are obtained based on the resultant band-pass filtered spectra. Such369
deviations at the shallow depths does not exist between the model results of370
Hm0 and Tm02 and the measurements. Comparing with the measurements,371
the model fairly reasonably predicts these second-order bulk statistics both372
in plunging and spilling dominated random breaking cases.373
As waves propagate from deep into shallower depths, crests and troughs374
become sharper and wider, respectively. Furthermore, waves pitch forward,375
and in the surf zone, the waveform becomes similar to a sawtoothed form.376
19
Normalized wave skewness= η3/(η2)3/2, and asymmetry= H(η)3/(η2)3/2 (where377
H denotes the Hilbert transform of the signal), are the statistical third-order378
moments characterizing these nonlinear features of a wave shape (Elgar &379
Guza, 1985; Mase & Kirby, 1992). Skewness and Asymmetry are the statisti-380
cal measures of asymmetry about horizontal and vertical planes, respectively.381
These third-order moments are potentially useful for sediment transport and382
morphology calculations. The bottom row of Figure 19 shows the cross-383
shore variation of the predicted third-order bulk statistics from outside the384
surf zone to the swash region. Comparing with the measurements, the model385
accurately captures the nonlinear effects, including the energy transfer due386
to triad nonlinear interaction, in the entire water depths, using as few as 4387
σ levels.388
3.2.3. Time-averaged velocity and k389
Although the only available data from Bowen & Kirby (1994) and Mase &390
Kirby (1992) are the free surface time series at different cross-shore locations,391
the predicted time-averaged velocity and k fields are presented and compared392
with those of regular breaking waves.393
Figure 20 shows the spatial distribution of the time-averaged velocity394
field using 4 and 8 σ levels for MK2. The normalized undertow current395
for the irregular wave cases have smaller magnitude than that for regular396
wave cases TK1 and TK2 with the same vertical structures within the surf397
zone. This is consistent with the measurements of Ting (2001) which has the398
20
similar incident wave conditions and experimental set-up compared with the399
simulated irregular breaking waves on a planner beach in the present study.400
In addition, the results with 4 σ levels have a nearly constant curvature at401
lower depths as oppose to the results with 8 levels where the curvature of the402
return current decreases at lower depths.403
Ting (2001) observed that the mean of the highest one-third wave-averaged404
k values in his irregular waves in the middle surf zone was about the same405
as k in a regular wave case TK1, where deep-water wave height to wave-406
length ratio of those two cases was on the same order. Here, the normalized407
k values are at the same order or even larger than those in regular breaking408
cases in the middle and inner surf zone. In the outer surf zone, however,409
the normalized k values are smaller than those under regular breaking cases.410
Although the k values decrease near the bottom in the outer surf zone similar411
to regular breaking cases, they have small vertical and cross-shore variations412
in the inner surf zone.413
4. Depth-limited breaking waves on a barred beach414
In this section, we use the data set of Scott et al. (2004), including a415
regular breaking case (hereafter referred as S1) and irregular breaking case416
(hereafter referred as S2), in order to examine the model predictions of free417
surface evolution as well as breaking-induced velocity and turbulence fields418
in depth-limited breaking waves on a barred beach. The experiment was con-419
ducted in the large wave flume at Oregon State University, approximately420
21
104m long, 3.7m wide, and 4.6m deep. The bathymetry was designed to421
approximate the bar geometry for the averaged profile observed on October422
11, 1994, of the DUCK94 field experiment at a 1:3 scale. The velocity mea-423
surements were carried out at 7 cross-shore locations using Acoustic Doppler424
Velocimeters (ADVs) sampling at 50 Hz. Figure 22 sketches the experimental425
layout and the cross-shore locations of the available free-surface and veloc-426
ity measurements. The regular case S1 is used by Jacobsen et al. (2014) to427
validate their 2D VOF-based model using RANS equations with k − ω tur-428
bulence closure. Here, both regular and irregular cases are considered; the429
corresponding results are given in §4.1 and §4.2 respectively. For both cases,430
a uniform grid of ∆x = 0.15m is used in the horizontal direction. Vertical431
resolutions of 4 and 8 σ levels are used. The right end of the numerical do-432
main is extended beyond the maximum run-up, and the wetting/drying cells433
are treated by setting Dmin = 0.001m for both S1 and S2.434
4.1. Regular breaking waves435
Table 3 summarizes the incident wave conditions for S1. The cross-shore436
location of the numerical wavemaker is set to be as the initial position of437
the physical wavemaker. The measured free surface and velocities deter-438
mined from linear theory are constructed at the wavemaker using the first439
10 Fourier components of the measured free surface time series in front of440
the wavemaker. In this section, 〈 〉 and ( ) refer to phase and time averaging441
over five subsequent waves after the results reach the quasi-steady state, re-442
22
Table 3: Input parameters for the simulated depth-limited regular breaking waves on abarred beach. Here, H0 and L0 are the deep water wave height and wave length calculatedusing linear theory, (kH)0 is the corresponding deep water wave steepness of the generatedwave, ξ0 = s/
√H0/L0 is the self similarity parameter, and s is the averaged slope before
the bar, assumed as s ∼ 1/12. For the irregular wave case S2, H = Hs0 is the deep-watercharacteristic wave height, T = Tp and k = kp, where p refers to the peak frequency ofthe incident waves.
Case no. H0 T (kH)0 ξ0 breaking(m) (s) type
S1 0.64 4.0 0.148 0.52 plungingS2 0.59 4.0 0.136 0.54 plunging
spectively. The corresponding measured averaged variables were calculated443
by phase averaging over 150 successive waves and ensemble averaging over444
at least 8 realizations.445
The mean sea level is defined as h = d+η, where d is the still water depth446
and η is the wave set-down/set-up. Here, x = 0 is the cross-shore location447
of the wavemaker location. The regular waves were observed to plunge at448
x = 53m.449
4.1.1. Time-dependent free surface evolution450
Figure 23 shows the cross-shore distribution of the wave height H =451
〈η〉max−〈η〉min as well as mean water level, η in the primary shoaling region452
up to the top of the bar (x < 52.8m), the top of the bar (52.8m< x < 56.5m),453
the shoreward face of the bar (56.5m< x < 60m), and the secondary shoaling454
region after the bar (x > 60m) for the regular case S1. The underprediction455
of the wave height near the breaking point is similar to that in TK1 as456
shown in Figure 2(a). Compared with measurements, wave height decay457
23
in the breaking region and shoreward face of the bar (53m< x < 60m) is458
captured reasonably well. In the secondary shoaling region after the bar459
(x > 60m), the overshoot of the wave height is not captured, as also seen in460
the VOF-based simulation of Jacobsen et al. (2014, Figure 4A). The mean461
water level is accurately resolved from deep water up to the swash zone, as462
opposed to the VOF-based simulation of Jacobsen et al. (2014, Figure 4B)463
which overpredicts wave set-up after the bar.464
Figure 24 shows the phase-averaged water surface elevations at different465
cross-shore locations before and after the bar for S1. Although the time466
evolution of the free surface elevations are comparable with the measurements467
at all cross-shore locations, the crest is underpredicted near the break-point468
as shown in panel (c) and after the bar as shown in panels (f) and (g). The469
secondary peak in the measured phase-averaged free surface elevations at470
x = 69.3m is also visible in the predicted results, while its crest elevation is471
underpredicted by the model. This secondary peak is due to the generation472
of the higher harmonics on top of the bar propagating with different phase473
speed than the primary wave. The predicted cross-shore location of the474
initial break point is slightly seaward compared with the measurements as in475
TK1, regardless of the different vertical resolutions. In both cases, the model476
captured the free surface evolution, wave height decay rate, crest and trough477
elevations, as well as wave set-up reasonably well using as few as 4 σ levels.478
24
4.1.2. Time-averaged velocity and k479
Figure 25 shows the spatial distribution of the time-averaged velocity field480
using different vertical resolutions for S1. To obtain the Eulerian mean ve-481
locities, the model results in the σ-coordinate system first were interpolated482
onto a fixed vertical mesh at each cross-shore location using linear interpo-483
lation, and then time averaging was performed. As in TK1, the predicted484
return current using 4 σ levels shown in 25(a) has not detached from the485
bed shoreward of the breaking point, as opposed to the simulation with 8 σ486
levels. The results of the simulations with different vertical resolutions have487
approximately the same structure after the breaking point, where the pre-488
dicted undertow current using 8 σ levels has larger magnitude in the entire489
surf zone. The curvature of the undertow profile has strong spatial varia-490
tions near the break points as shown in Figure 26(c), where the amount of491
curvature of the undertow profile at x = 48.0m (red lines) considerably de-492
creases compared with that at x = 51.0m (black lines). This is due to the493
detachment of the undertow current from the bed, forming negative slopes at494
seaward of the break point. Figure 26(c) also shows that the model predicts495
breaking seaward of the measured break point. Finally, the measured under-496
tow profiles at two different longshore locations (shown by open and solid497
circles) reveal that the time-averaged velocity field has strong variation in498
the spanwise direction close to the break point; the 3D effects are absent in499
our 2D simulation. Compared with the measured undertow profiles (Figure500
26), the undertow current is resolved on top of and after the bar using as few501
25
as 4 σ levels.502
Figure 27 shows the spatial distribution of k using different vertical res-503
olutions for S1. The values of the normalized time-averaged k,√k/gh, are504
similar to those in TK1 and TK2 in the outer surf zone. Figure 28 shows the505
predicted k profiles at the different cross-shore locations before, on the top of,506
and after the bar together with the corresponding measurements. Compared507
with the measurements, it is seen that the model predicts fairly reasonably508
the cross-shore variation of the breaking-induced turbulence using 4 σ levels,509
with the large k levels across the breaker bar, where the waves are breaking,510
and the subsequent decay of k level on the seaward face as well as after the511
bar.512
4.2. Irregular breaking waves513
The random waves of S2 were generated based on a TMA spectrum with a514
width parameter γ = 20 to generate the initial condition at the wavemaker.515
Table 3 summarizes the incident wave conditions for S2. The cross-shore516
location of the numerical wavemaker is set to be as the initial position of the517
physical wavemaker. The measured free surface and velocities determined518
from linear theory are constructed at the wavemaker using the first 2000519
Fourier components of the measured free surface time series in front of the520
wavemaker. In this section, ( ) refers to long-time averaging over several521
minutes, more than 250 waves. The first 2500 data points were ignored both522
in the model and results and the corresponding experiment.523
26
The mean sea level is defined as h = d + η, where d is the still water524
depth and η is the wave set-down/set-up. Here, x = 0 is the cross-shore525
location of the wavemaker location. The random waves were observed to be526
both plunging and spilling as far offshore as x = 42m.527
4.2.1. Power spectra evolution and integral breaking-induced dissipation528
Here, we examine the model prediction of the integral breaking-induced529
dissipation compared with the corresponding measurements by looking at530
the evolution of the power spectral density, S(f), across a fixed bar.531
Figure 29 shows the variation of computed S(f) using 4 and 8 σ lev-532
els for the random breaking case S2 as well as the corresponding measured533
S(f). The measured signals were split into 8196 data points segments. Each534
segment multiplied by a cosine-taper window with the taper ratio of 0.05535
to reduce the end effects. The measured spectrum is obtained by ensemble536
averaging over the computed spectra of 7 segments and then band averag-537
ing over the 5 neighboring bands. Thus the resultant averaged spectra have538
70 degrees of freedom. The sampling rate was 50 Hz (fNyq = 25Hz). The539
spectrum resolution is ∆f = 0.03Hz. The computed spectrum is obtained540
in a similar way, with the same spectral resolution and degrees of freedom.541
Panels (a),(b), and (c) show the S(f) in the shoaling zone before the break542
point x = 53m. The decrease of energy at the dominant peak frequency543
and increase of energy at higher and lower harmonics before the breaking544
region due to the nonlinear interaction, shown at panel (c), as well as the545
27
decrease of energy at the dominant peak frequency and higher frequency546
range across the bar, shown in panel (d), are captured by the model using 4547
σ levels. However, the energy at low-frequency range is overpredicted while548
the energy at the second harmonic is underpredicted across and after the549
bar. No wave absorption at the wavemaker exists both in the simulation550
and the experiment, and thus the reflected long waves from the bar and the551
beach face are reflected back in the domain as in the experiment. The more552
pronounced predicted energy at this frequency range (f/fp ≈ 0.5) comparing553
with the measurements may be due to the inherent difference between the554
numerical wavemaker and that in the experiment and the absence of lateral555
side walls effects in the present 2D simulation. The underprediction of the556
second harmonics across the bar is unresolved.557
4.2.2. Wave statistics558
Figure 30(a) shows the cross-shore variations of the model predictions of559
η, Hm0 , Tm02 , normalized wave skewness, and normalized wave asymmetry560
using 4 and 8 σ levels as well as the corresponding measured values for the561
random breaking case S2. These bulk statistics are calculated as explained562
in §3.2.1. Comparing with the measurements, the model fairly reasonably563
predicts the wave set-down/set-up as well as the second- and third-order bulk564
statistics for S2 using 4 σ levels. As in the regular case S1 (Figure 23a), the565
wave height after the bar, x > 60m, is underpredicted.566
28
4.2.3. Time-averaged velocity and k field567
Figure 31 shows the spatial distribution of the time-averaged velocity568
field using different vertical resolutions of 4 and 8 levels for S2. The Eulerian569
mean velocities were obtained as described before. The predicted undertow570
current using 4 and 8 σ levels have approximately the same structure and571
magnitude in the surf zone, and have the smaller magnitude compared with572
those under the regular case S1. Comparing the results with the measured573
undertow profiles shown in Figure 32, the undertow current is reasonably574
well captured across the bar and trough using as few as 4 σ levels, with575
smaller amount of curvature at lower depths which is partially because of the576
underprediction of the k and as a result the unerprediction of the turbulent577
eddy viscosity at those depths, as explained in §3.1.2.578
Figure 33 shows the spatial distribution of the time-averaged k field us-579
ing different vertical resolutions for S2. The values of the normalized time-580
averaged k,√k/gh, are smaller than those in the regular case S1 in the entire581
surf zone, having the same structure near the bar and the steep beach. Figure582
34 shows the predicted time-averaged k profiles at the different cross-shore583
locations before, on the top of, and after the bar together with the corre-584
sponding measurements. Compared with the measurements, it is seen that585
using 4 σ levels the model predicts fairly reasonably the cross-shore variation586
of the breaking-induced turbulence as in the regular case S1.587
29
5. Steepness-limited unsteady breaking waves588
The data sets of Rapp & Melville (1990) and Tian et al. (2012) are con-589
sidered to study the model capability and accuracy for breaking-induced590
processes in steepness-limited unsteady breaking waves. Here, the model591
results for the two unsteady plunging breakers of Rapp & Melville (1990),592
hereafter referred as RM1 and RM2, in an intermediate depth regime with593
kcd ≈ 1.9 and one of the plunging cases of Tian et al. (2012), hereafter re-594
ferred as T1, in a deep water regime with kcd ≈ 6.9 are presented, where kc595
is the wave number of the center frequency wave of the input packet defined596
below. The evolution of the free surface, mean velocity field and large mean597
vortex under isolated breaking case RM1 are compared to the corresponding598
measurements and the results of the VOF-based simulation of Derakhti &599
Kirby (2014b). Integral breaking-induced energy dissipation under an iso-600
lated steepness-limited unsteady breaking wave is examined for RM2. In601
addition, the power spectral density evolution as well as integral breaking-602
induced energy dissipation under multiple steepness-limited unsteady break-603
ing waves are examined for T1.604
In both experiments, breaking waves were generated using the dispersive605
focusing technique, in which an input packet propagates over an constant606
depth and breaks at a predefined time, tb, and location, xb. The input wave607
packet was composed of N sinusoidal components of steepness aiki where the608
ai and ki are the amplitude and wave number of the ith component. Based609
on linear superposition and by imposing that the maximum 〈η〉 occurs at xb610
30
and tb, the total surface displacement at the incident wave boundary can be611
obtained as (Rapp & Melville, 1990, §2.3)612
〈η〉(0, t) =N∑i=1
ai cos[2πfi(t− tb) + kixb], (1)
where fi is the frequency of the ith component. The discrete frequencies fi613
were uniformly spaced over the band ∆f = fN − f1 with a central frequency614
defined by fc = 12(fN − f1). Different global steepnesses S =
∑Ni=1 aiki and615
normalized band-widths ∆f/fc lead to spilling or plunging breaking, where616
increasing S and/or decreasing ∆f/fc increases the breaking intensity (See617
Drazen et al. (2008) for more details). In the numerical wavemaker, free sur-618
face and velocities of each component are calculated using linear theory and619
then superimposed at x = 0. Sponge levels are used at the right boundary620
to minimize reflected waves. The input wave parameters for different cases621
are summarized in table 4.622
The normalized time and locations are defined as623
x∗ =x− xobLc
, z∗ =z
Lc
, t∗ =t− tobTc
, (2)
where Tc and Lc are the period and wavelength of the center frequency wave624
of the input packet, respectively. Here, tob and xob are the time and location625
at which the forward jet hits the free surface, obtained from corresponding626
VOF simulations of Derakhti & Kirby (2015).627
31
Table 4: Input parameters for the simulated focused wave packets. d is the still waterdepth, S =
∑Ni=1 aiki is the global steepness, N is the number of components in the
packet, aiki is the component steepness which is the same for the all components, and thediscrete frequencies fi were uniformly spaced over the band ∆f = fN − f1 with a centralfrequency defined by fc = 1
2 (fN − f1).
Case no. d S fc ∆f/fc N breaking(m) (1/s) type
RM1 0.60 0.352 0.88 0.73 32 plungingRM2 0.60 0.388 0.88 0.73 32 plungingT1 0.62 0.576 1.70 0.824 128 plunging
5.1. Time-dependent free surface evolution628
Figure 35 shows the free surface evolution in the breaking region for RM1629
using 8 σ levels. Figure 36 shows the free surface time series at locations630
before and after the break point, showing that the model captures the free631
surface evolution up to the break point fairly accurately. The overall wave632
height decay is also predicted reasonably well. However, the sudden drop of633
the crest during active breaking is not resolved.634
Figure 37 shows the water surface elevations at different x locations for635
T1 using 8 σ levels. Nearly all the input wave components are in the deep636
water regime (d/Li > 0.5), and thus the packet is highly dispersive. Multiple637
breaking was observed in the experiment between x∗ ≈ −1 and x∗ ≈ 1, where638
x∗ = 0 is the x location of the main breaking event in the packet. The model639
captures the packet propagation and evolution accurately. The focusing of640
dispersive waves before the break point can be seen at panels (a) through641
(c) with decrease in the number of waves and increase of the maximum crest642
32
elevation. Downstream of the breaking region (Figure 37e and f), the results643
indicate that the wave height decay due to multiple unsteady breaking events,644
as well as dispersive properties of the packet, are captured by the model645
reasonably well.646
5.2. Integral Breaking-Induced Dissipation647
In this section, the predicted integral breaking-induced dissipation is com-648
pared to the corresponding measurements by looking at the evolution of the649
time-integrated energy density, ρgη2, as well as the power spectral density.650
In this section, ( ) refers to long-time integration over the entire wave packet.651
Strictly speaking, ρgη2 is twice the time-integrated potential energy den-652
sity, Ep, and, to a good approximation, can be considered as the time-653
integrated total energy density far from the breaking region. By choosing654
an appropriate characteristic group velocity, Cgρgη2 is then used as an es-655
timation of the time-integrated total horizontal energy flux, F . Thus, the656
spatial variation of ρgη2 is related to total breaking-induced dissipation for657
unsteady breaking waves, as explained by Derakhti & Kirby (2015) in detail.658
Figure 38 shows the variation of η2/η21 for the intermediate depth unsteady659
breaking case, RM2, using different horizontal and vertical resolutions. The660
predicted integral dissipation is underestimated comparing with the mea-661
surements. In addition, the predicted decay of Ep occurs at a larger down662
wave distance compared with the measurements, and the sudden drop of the663
potential energy density is not resolved.664
33
Here, the entire dissipation is imposed by the shock-capturing TVD scheme665
in these cases. In other words, the turbulence model has not been triggered,666
and νt is approximately zero. It is well known that the numerical dissipa-667
tion applied by TVD schemes decreases as the grid resolution increases. In668
breaking waves, the large gradient in a velocity field occurs near the sharp669
wave front and in the horizontal direction. As expected, by decreasing the670
horizontal resolution from ∆x = 23mm to ∆x = 10mm the total decay of Ep671
becomes smaller, whereas the associated change in Ep due to further decrease672
of ∆x from 10 mm to 5 mm is negligibly small. Increasing the vertical reso-673
lution, on the other hand, improves the results. Similar behavior is observed674
in other cases (not shown).675
Figure 39 shows the evolution of different spectral components in the wave676
packet for T1, and the corresponding measurements of Tian et al. (2012). The677
measured spectrum is obtained by ensemble averaging over 5 runs and then678
band averaging over three neighboring bands (30 degrees of freedom) with679
a spectral resolution of ∆f = 0.075Hz, where the signal length is 40 s, and680
the sampling rate is 100 Hz. The computed spectrum is based on a single681
realization with the same length and sampling rate. In general, the energy682
of the high frequency (f/fc > 2) part of the spectrum is underestimated683
due to a relatively coarse vertical resolution of the model which can not684
resolved fast decay of short-waves orbital velocities with depth. The nonlinear685
energy transfer into low-frequency components (f/fc < 0.5), however, is686
fairly reasonably resolved. Energy is dissipated mostly in the frequency range687
34
0.75 < f/fc < 1.5, as shown in panels (e) and (f). Close to the break688
point, the model does not capture the sudden dissipation of energy, especially689
for larger frequencies (Figure 39c). The predicted spectrum becomes more690
similar to the measured spectrum as the packet propagates away from the691
breaking region.692
5.3. Velocity field693
Comprehensive experimental work by Rapp & Melville (1990) and Drazen694
& Melville (2009) has revealed the main characteristics of the ensemble-695
averaged flow field under unsteady breaking waves, especially after active696
breaking. Rapp & Melville (1990) measured the velocity field using LDV697
at seven elevations and seven x locations in the breaking region. Figure698
40 shows the normalized horizontal and vertical velocities at x∗ = 0.60,699
z∗ = −0.025 for RM1 using 10 σ levels versus the corresponding unfiltered700
measured ensemble-averaged signals. After breaking, the larger velocities701
compared with the measurements also demonstrates the underprediction of702
the breaking-induced dissipation shown in Figure 38.703
The ensemble-averaged velocity field can be decomposed into704
〈u〉 = uw + ufw + uc, (3)
where uw is the orbital velocity of the surface waves, ufw is the velocity of the705
forced long-waves induced by breaking, and uc is the current stemming from706
the momentum loss during the breaking and/or Stokes drift. The rest of the707
35
available measured velocity signals are low-pass filtered using the threshold708
frequency of 0.3 Hz, to remove the surface waves as in Rapp & Melville709
(1990), where the frequency range of the input surface waves is 0.56 < fi <710
1.20. Figure 41 shows the low-pass filtered results and the corresponding711
measurements for RM1 at x∗ = 0.15 and x∗ = 0.60, from very close to the712
free surface to z∗ = −0.15 (≈ z = −d/2). The smaller low-passed filtered713
velocity field is due to the smaller wave dissipation and smaller wave forcing,714
predicted by the model.715
The mean current can be calculated by time averaging of the ensemble-716
averaged velocity signal,717
uc = u =1
t∗2 − t∗1
∫ t∗2
t∗1
〈u〉 dt∗, (4)
where t∗1 and t∗2 cover the entire wave packet. During time integration for each718
grid point, when the point is above the free surface the velocity signal is zero.719
Figure 42 shows the spatial distribution of the normalized mean current and720
its horizontal-averaged between x∗ = 0 and 1.5, as well as the normalized721
horizontal-averaged mass flux below the depth z∗, M∗(z∗) =∫ z∗
z∗1u∗cdz
∗ where722
z∗1 = −0.31 is the bottom elevation, for RM1 using 8 σ levels (top panels)723
together with the LES/VOF results by Derakhti & Kirby (2014b) (bottom724
panels). The positive current near the surface, the return negative current725
at lower depths and the two distinct circulation cells are captured by the726
model as in the LES/VOF results. Comparing with the measurements of727
36
(Rapp & Melville, 1990, Figure 43) and the LES/VOF simulation, we can728
see that the model generated a large mean vortex with relatively stronger729
velocity field. We believe this is due to the absence of an enhanced eddy730
viscosity that would be present as a result of the turbulence, which was731
not captured by NHWAVE in unsteady breaking cases. In addition, the732
model predicts relatively larger cells than those predicted by the LES/VOF733
simulation, especially in the x direction. The predicted patch of persistent734
vorticity (not shown) is consistent with Drazen & Melville (2009, Figure 4)735
and the LES/VOF simulation of Derakhti & Kirby (2014b, Figure 4.16),736
having larger vorticity values due to underestimation of effective viscosity in737
the absence of turbulence.738
6. Conclusions739
In this paper, we examined wave-breaking predictions ranging from shallow-740
to deep-water conditions using a surface-following, shock-capturing 3D non-741
hydrostatic model, NHWAVE (Ma et al., 2012), comparing results both with742
corresponding experiments and with outcomes of a VOF/Navier-Stokes solver743
(Ma et al., 2011; Derakhti & Kirby, 2014a,b). The new version of NHWAVE744
has been described in Derakhti et al. (2015), including the new governing745
equations and exact surface and bottom boundary conditions. We consid-746
ered regular and irregular depth-limited breaking waves on planar and barred747
beaches as well as steepness-limited unsteady breaking waves in intermediate748
and deep depths. The same equations and numerical methods are used for749
37
the various depth regimes and involve no ad-hoc treatment. Vertical grid750
resolution in all simulated cases is at least an order of magnitude coarser751
than that of typical VOF-based simulations. The main conclusions can be752
categorized as follows.753
(a) Depth-limited breaking waves: using as few as 4 σ levels, the model754
was shown to accurately predict depth-limited breaking wave properties in755
terms of (1) time-dependent free-surface and mean velocity field evolution,756
(2) integral breaking-induced dissipation, (3) second- and third-order bulk757
statistics, and (4) breaking-induced organized motion both on a planar and758
barred beaches. In addition, the model is shown to predict k distributions759
under troughs as accurate as those predicted by typical VOF-based simula-760
tions without bubble effects. As it was explained by Derakhti et al. (2015),761
the new boundary conditions significantly improve the predicted velocity762
and turbulence fields under depth-limited breaking waves compared with the763
commonly used simplified stress boundary conditions, ignoring the effects764
of surface and bottom slopes in the transformation of stress terms. The k765
prediction above the troughs may be further improved by replacing the zero766
gradient boundary condition for k and/or the zero-stress tangential stress767
boundary with a physics-based model such as the model proposed by Broc-768
chini & Peregrine (2001); Brocchini (2002). Under strong plunging breakers,769
the rapid advection of high k to lower depths can not captured by the model770
due to the unresolved jet impact and subsequent splash processes. It was771
found that this turbulence underprediction, and thus the underprediction of772
38
the turbulent eddy viscosity, can not be improved by increasing the number773
of σ levels. As a result, the amount of the curvature of undertow profiles774
are overpredicted in the events where the breaking is characterized as strong775
plunging.776
(b) Steepness-limited breaking waves: it was shown that all the dissipa-777
tion was imposed indirectly by only the TVD shock-capturing scheme, and778
the turbulence model had not been triggered. Although the absence of tur-779
bulence in deep water breaking waves predictions led to the underestimation780
of the total breaking-induced dissipation, and, thus, the overprediction of the781
velocity and vorticity field in the breaking region, the model was shown to782
predict (1) the dispersive and nonlinear properties of different wave packet783
components before and after the break point, (2) the overall wave height784
decay and spectral evolutions, and (3) the structures of the mean velocity785
and vorticity fields including large breaking-induced coherent vortices. The786
near-surface turbulence model for whitecap events, e.g., the model proposed787
by Brocchini (2002) to set boundary condition for k, is needed to provide788
sufficient k levels during active breaking, with which the model will produce789
the turbulence field, leading to an enhance eddy viscosity and an appropriate790
amount of breaking-induced dissipation in the breaking region.791
Acknowledgments792
The authors gratefully thank Zhigang Tian, Marc Perlin, and Wooyoung793
Choi for providing the deep water laboratory data. This work was sup-794
39
ported by ONR, Littoral Geosciences and Optics Program (grant N00014-13-795
1-0124); NSF, Physical Oceanography Program (grant OCE-1435147); and796
through the use of computational resources provided by Information Tech-797
nologies at the University of Delaware.798
Ai, C., Ding, W. & Jin, S. 2014 A general boundary-fitted 3d non-799
hydrostatic model for nonlinear focusing wave groups. Ocean Eng. 89,800
134–145.801
Bowen, G. D. & Kirby, J. T. 1994 Shoaling and breaking random waves802
on a 1: 35 laboratory beach. Tech. Rep.. Tech. Rep. CACR-94-14.803
Bradford, S. F. 2000 Numerical simulation of surf zone dynamics. J.804
Waterway, Port, Coastal, and Ocean Eng. 126, 1–13.805
Bradford, S. F. 2011 Nonhydrostatic model for surf zone simulation. J.806
Waterway, Port, Coastal, and Ocean Eng. 137, 163–174.807
Bradford, S. F. 2012 Improving the efficiency and accuracy of a nonhy-808
drostatic surf zone model. Coastal Eng. 65, 1–10.809
Brocchini, M. 2002 Free surface boundary conditions at a bubbly/weakly810
splashing air–water interface. Phys. Fluids 14, 1834–1840.811
Brocchini, M. & Peregrine, D. H. 2001 The dynamics of strong tur-812
bulence at free surfaces. part 2. free-surface boundary conditions. J. Fluid813
Mech. 449, 255–290.814
40
Christensen, E. D. 2006 Large eddy simulation of spilling and plunging815
breakers. Coastal Eng. 53, 463–485.816
Derakhti, M. & Kirby, J. T. 2014a Bubble entrainment and liquid-817
bubble interaction under unsteady breaking waves. J. Fluid Mech. 761,818
464–506.819
Derakhti, M. & Kirby, J. T. 2014b Bubble entrainment and liquid-820
bubble interaction under unsteady breaking waves. Res. Rep. CACR-14-821
06, Center for Applied Coastal Research, University of Delaware .822
Derakhti, M. & Kirby, J. T. 2015 Energy and momentum flux under823
unsteady breaking waves. submitted to J. Fluid Mech. .824
Derakhti, M., Kirby, J. T., Shi, F. & Ma, G. 2015 New version of825
NHWAVE: Governing equations and exact boundary conditions. Ocean826
Modelling submitted.827
Drazen, D. A. & Melville, W. K. 2009 Turbulence and mixing in un-828
steady breaking surface waves. J. Fluid Mech. 628, 85.829
Drazen, D. A., Melville, W. K. & Lenain, L. 2008 Inertial scaling of830
dissipation in unsteady breaking waves. J. Fluid Mech. 611, 307–332.831
Elgar, S. & Guza, R.T. 1985 Observations of bispectra of shoaling surface832
gravity waves. J. Fluid Mech. 161, 425–448.833
41
Garcez Faria, A. F., Thornton, E. B., Lippmann, T. C. & Stan-834
ton, T. P. 2000 Undertow over a barred beach. J. Geophys. Res. 105,835
16999–17010.836
Jacobsen, N. G, Fredsoe, J. & Jensen, J. H 2014 Formation and de-837
velopment of a breaker bar under regular waves. part 1: Model description838
and hydrodynamics. Coastal Eng. 88, 182–193.839
Kaihatu, J. M., Veeramony, J., Edwards, K. L. & Kirby, J. T. 2007840
Asymptotic behavior of frequency and wave number spectra of nearshore841
shoaling and breaking waves. J. Geophys. Res.: Oceans 112, C06016.842
Lakehal, D. & Liovic, P. 2011 Turbulence structure and interaction with843
steep breaking waves. J. Fluid Mech. 674, 522–577.844
Lin, P. & Liu, P.L.-F 1998 A numerical study of breaking waves in the845
surf zone. J. Fluid Mech. 359, 239–264.846
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans.847
Roy. Soc. London, A 245, 535–581.848
Longuet-Higgins, M. S. 1970 Longshore currents generated by obliquely849
incident sea waves: 1. J. Geophys. Res. 75, 6778–6789.850
Lubin, P. & Glockner, S. 2015 Numerical simulations of three-851
dimensional plunging breaking waves: generation and evolution of aerated852
vortex filaments. J. Fluid Mech. 767, 364–393.853
42
Ma, G., Chou, Y. & Shi, F. 2014a A wave-resolving model for nearshore854
suspended sediment transport. Ocean Mod. 77, 33–49.855
Ma, G., Kirby, J. T. & Shi, F. 2013a Numerical simulation of tsunami856
waves generated by deformable submarine landslides. Ocean Mod. 69, 146–857
165.858
Ma, G., Kirby, J. T., Su, S., Figlus, J. & Shi, F. 2013b Numerical859
study of turbulence and wave damping induced by vegetation canopies.860
Coastal Eng. 80, 68–78.861
Ma, G., Shi, F., Hsiao, S. & Wu, Y. 2014b Non-hydrostatic modeling862
of wave interactions with porous structures. Coastal Eng. 91, 84–98.863
Ma, G., Shi, F. & Kirby, J. T. 2011 A polydisperse two-fluid model for864
surf zone bubble simulation. J. Geophys. Res.: Oceans 116, C05010.865
Ma, G., Shi, F. & Kirby, J. T. 2012 Shock-capturing non-hydrostatic866
model for fully dispersive surface wave processes. Ocean Mod. 43, 22–35.867
Mase, H. & Kirby, J. T. 1992 Hybrid frequency-domain KdV equation868
for random wave transformation. In Proc. 23d Int. Conf. Coastal Eng., pp.869
474–487. Venice.870
Phillips, OM 1977 The dynamics of the upper ocean. Cambridge University871
Press, London.872
43
Pizzo, N. E. & Melville, W K. 2013 Vortex generation by deep-water873
breaking waves. J. Fluid Mech. 734, 198–218.874
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-875
water breaking waves. Phil. Trans. Roy. Soc. A, 331, 735–800.876
Scott, C. P, Cox, D. T, Shin, S. & Clayton, N. 2004 Estimates of877
surf zone turbulence in a large-scale laboratory flume. In Proc. 29th Int.878
Conf. Coastal Eng., pp. 379–391.879
Shirkavand, A. & Badiei, P. 2014 The application of a godunov-type880
shock capturing scheme for the simulation of waves from deep water up to881
the swash zone. Coastal Eng. 94, 1–9.882
Smit, P., Janssen, T., Holthuijsen, L. & Smith, J. 2014 Non-883
hydrostatic modeling of surf zone wave dynamics. Coastal Eng. 83, 36–48.884
Smit, P., Zijlema, M. & Stelling, G. 2013 Depth-induced wave break-885
ing in a non-hydrostatic, near-shore wave model. Coastal Eng. 76, 1–16.886
Svendsen, I. A. 1984 Mass flux and undertow in a surf zone. Coastal Eng.887
8, 347–365.888
Tappin, D. R., Grilli, S. T., Harris, J. C., Geller, R. J., Master-889
lark, T., Kirby, J. T., Shi, F., Ma, G., Thingbaijam, K. K. S. &890
Mai, P. M. 2014 Did a submarine landslide contribute to the 2011 tohoku891
tsunami? Marine Geology 357, 344–361.892
44
Tian, Z., Perlin, M. & Choi, W. 2012 An eddy viscosity model for893
two-dimensional breaking waves and its validation with laboratory exper-894
iments. Phys. of Fluids 24, 036601.895
Ting, F. C. K. 2001 Laboratory study of wave and turbulence velocities in896
a broad-banded irregular wave surf zone. Coastal Eng. 43, 183–208.897
Ting, F. C. K. & Kirby, J. T. 1994 Observation of undertow and turbu-898
lence in a laboratory surf zone. Coastal Eng. 24, 51–80.899
Toro, E. F. 2009 Riemann solvers and numerical methods for fluid dynam-900
ics: a practical introduction. Springer.901
Trowbridge, J. & Madsen, O. S. 1984 Turbulent wave boundary layers:902
2. second-order theory and mass transport. J. Geophys. Res. 89, 7999–903
8007.904
Watanabe, Y., Saeki, H. & Hosking, R. J. 2005 Three-dimensional905
vortex structures under breaking waves. J. Fluid Mech. 545, 291–328.906
Wiegel, RL 1960 A presentation of cnoidal wave theory for practical ap-907
plication. J. Fluid Mech. 7, 273–286.908
Yakhot, V. Orszag, S., Thangam, S, Gatski, T. & Speziale, C.909
1992 Development of turbulence models for shear flows by a double expan-910
sion technique. Phys. Fluids 4, 1510–1520.911
45
Young, C. & Wu, C. H. 2010 Nonhydrostatic modeling of nonlinear deep-912
water wave groups. J. Eng. Mech. 136, 155–167.913
Zhou, Z., Sangermano, J., Hsu, T. & Ting, F. C. K. 2014 A numerical914
investigation of wave-breaking-induced turbulent coherent structure under915
a solitary wave. J. Geophys. Res.: Oceans 119, 6952–6973.916
Zijlema, M., Stelling, G. & Smit, P. 2011 SWASH: An operational917
public domain code for simulating wave fields and rapidly varied flows in918
coastal waters. Coastal Eng. 58, 992–1012.919
46
0.38md0 = 0.4m
1 : 35
SWL
Wave Maker
x
z
Figure 1: Experimental layout of Ting & Kirby (1994). Vertical solid lines: the cross-shorelocations of the velocity measurements for TK1. Vertical dashed lines: the cross-shorelocations of the velocity measurements for TK2.
-2.5 0 2.5 5 7.5 10 12.5
-8
0
8
16
η
〈η〉max
− η
〈η〉min
− η
(a) (A)
x (m)-2.5 0 2.5 5 7.5 10 12.5
-8
0
8
16 (b)
η
〈η〉max
− η
〈η〉min
− η
x∗ (m)-2.5 0 2.5
(B)
Figure 2: Cross-shore distribution of crest and trough elevations as well as mean waterlevel for the surf zone (a,A) spilling breaking case TK1 and (b,B) plunging breaking caseTK2. Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels(dotted-dashed lines), 16 σ levels (solid lines) and the measurements of Ting & Kirby(1994) (circle markers). In panels (A) and (B), x∗ = x − xb represents the horizontaldistance from the break point.
47
〈η〉−
η(m
)
-5
0
5
10
15(a)
x∗ = -0.46m
(c)
x∗ = 0.88m
t/T0 0.2 0.4 0.6 0.8 1
〈η〉−
η(m
)
-5
0
5
10
15(b)
x∗ = 0.26m
t/T0 0.2 0.4 0.6 0.8 1
(d)
x∗ = 1.48m
Figure 3: Phase-averaged free surface elevations for the surf zone spilling breaking caseTK1 at different cross-shore locations before and after the initial break point x∗ = 0.Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines) and the measurement (thin red solid lines).
〈η〉−
η(m
)
-5
0
5
10
15 (a)
x∗ = -0.50m
(c)
x∗ = 0.55m
t/T0 0.2 0.4 0.6 0.8 1
〈η〉−
η(m
)
-5
0
5
10
15 (b)
x∗ = 0.00m
t/T0 0.2 0.4 0.6 0.8 1
(d)
x∗ = 1.00m
Figure 4: Phase-averaged free surface elevations for the surf zone plunging breaking caseTK2 at different cross-shore locations before and after the initial break point x∗ = 0.Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines) and the measurement (thin red solid lines).
48
(〈u〉−
u)/√gh
-0.4
0
0.4 (a)
x∗ = -7.67m
z∗ = 6.00cm
(〈u〉−
u)/√gh
-0.4
0
0.4 (b)
x∗ = -0.46m
z∗ = 4.80cm
(〈u〉−
u)/√gh
-0.4
0
0.4 (c)
x∗ = 0.26m
z∗ = 4.50cm
t/T0 0.2 0.4 0.6 0.8 1
(〈u〉−
u)/√gh
-0.4
0
0.4 (d)
x∗ = 0.88m
z∗ = 4.90cm
(e)
x∗ = 1.48m
z∗ = 5.20cm
(f)
x∗ = 2.09m
z∗ = 4.70cm
(g)
x∗ = 2.71m
z∗ = 4.90cm
t/T0 0.2 0.4 0.6 0.8 1
(h)
x∗ = 3.32m
z∗ = 4.70cm
Figure 5: Phase-averaged normalized horizontal velocities for the surf zone spilling break-ing case TK1 at about 5 cm above the bed (z∗ is the distance from the bed), at differentcross-shore locations before and after the initial break point x∗ = 0. Comparison betweenNHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines), 16 σlevels (thick solid lines) and measurements (thin red solid lines).
49
(〈u〉−
u)/√gh
-0.4
0
0.4 (a)
x∗ = -0.50m
z∗ = 4.90cm
(〈u〉−
u)/√gh
-0.4
0
0.4 (b)
x∗ = 0.00m
z∗ = 4.60cm
(〈u〉−
u)/√gh
-0.4
0
0.4 (c)
x∗ = 0.55m
z∗ = 5.20cm
t/T0 0.2 0.4 0.6 0.8 1
(〈u〉−
u)/√gh
-0.4
0
0.4 (d)
x∗ = 1.00m
z∗ = 4.80cm
(e)
x∗ = 1.50m
z∗ = 5.30cm
(f)
x∗ = 2.00m
z∗ = 4.60cm
t/T0 0.2 0.4 0.6 0.8 1
(g)
x∗ = 2.60m
z∗ = 4.90cm
Figure 6: Phase-averaged normalized horizontal velocities for the surf zone plunging break-ing case TK2 at about 5 cm above the bed (z∗ is the distance from the bed), at differentcross-shore locations before and after the initial break point x∗ = 0. Comparison betweenNHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines), 16 σlevels (thick solid lines) and measurements (thin red solid lines).
50
Figure 7: Time-averaged velocity field, u, for the surf zone spilling breaking case TK1.NHWAVE results with (a) 4 σ levels, (b) 8 σ levels, and (c) 16 σ levels. Dash lines showthe crest 〈η〉max and trough 〈η〉min elevations. Colors show u/
√gh.
51
(z−η)/h
-1
-0.5
0
(a)
x∗ = -7.67m
(z−η)/h
-1
-0.5
0
(b)
x∗ = -0.46m
(z−η)/h
-1
-0.5
0
(c)
x∗ = 0.26m
u/√
gh-0.25 0 0.25
(z−η)/h
-1
-0.5
0
(d)
x∗ = 0.88m
(e)
x∗ = 1.48m
(f)
x∗ = 2.09m
(g)
x∗ = 2.71m
u/√
gh-0.25 0 0.25
(h)
x∗ = 3.32m
Figure 8: Time-averaged normalized horizontal velocity (undertow) profiles for the surfzone spilling breaking case TK1 at different cross-shore locations before and after the initialbreak point, x∗ = 0. Comparison between NHWAVE results with 4 σ levels (dashed lines),8 σ levels (dotted-dashed lines), 16 σ levels (solid lines) and the measurements (circlemarkers).
52
(z−η)/h
-1
-0.5
0
(a)
x∗ = -0.50m
(z−η)/h
-1
-0.5
0
(b)
x∗ = 0.00m
(z−η)/h
-1
-0.5
0
(c)
x∗ = 0.55m
u/√
gh-0.25 0 0.25
(z−η)/h
-1
-0.5
0
x∗ = 1.00m
(d)
(e)
x∗ = 1.50m
(f)
x∗ = 2.00m
u/√
gh-0.25 0 0.25
x∗ = 2.60m
(g)
Figure 9: Time-averaged normalized horizontal velocity (undertow) profiles for the surfzone plunging breaking case TK2 at different cross-shore locations before and after theinitial break point, x∗ = 0. Comparison between NHWAVE results with 4 σ levels (dashedlines), 8 σ levels (dotted-dashed lines), 16 σ levels (solid lines) and the measurements (circlemarkers).
53
Figure 10: Snapshots of the turbulent kinetic energy, k(m2/s2), distribution for the surfzone spilling breaking case TK1. NHWAVE results with (a− e) 4 σ levels and (A−E) 8σ levels.
54
√
〈k〉/gh
0
0.05(a)
x∗ = -0.46m
√
〈k〉/gh
0
0.05(b)
x∗ = 0.26m
√
〈k〉/gh
0
0.1 (c)
x∗ = 0.88m
√
〈k〉/gh
0
0.1 (d)
x∗ = 1.48m
√
〈k〉/gh
0
0.1 (e)
x∗ = 2.09m
t/T0 0.2 0.4 0.6 0.8 1
√
〈k〉/gh
0
0.1 (f)
x∗ = 2.71m
(A)
x∗ = -0.46m
(B)
x∗ = 0.26m
(C)
x∗ = 0.88m
(D)
x∗ = 1.48m
(E)
x∗ = 2.09m
t/T0 0.2 0.4 0.6 0.8 1
(F )
x∗ = 2.71m
Figure 11: Phase-averaged k time series for the surf zone spilling breaking case TK1 at(a−f) ∼ 4 cm and (A−F ) ∼ 9 cm above the bed at different cross-shore locations beforeand after the initial break point, x∗ = 0. Comparison between NHWAVE results with 4σ levels (dashed lines), 8 σ levels (dotted-dashed lines), 16 σ levels (thick solid lines) andthe measurement (thin red solid lines)
55
√
〈k〉/gh
0
0.1(a)
x∗ = 0.00m
√
〈k〉/gh
0
0.1
0.2 (b)
x∗ = 0.55m
√
〈k〉/gh
0
0.1
0.2 (c)
x∗ = 1.00m
√
〈k〉/gh
0
0.1
0.2 (d)
x∗ = 1.50m
√
〈k〉/gh
0
0.1
0.2 (e)
x∗ = 2.00m
t/T0 0.2 0.4 0.6 0.8 1
√
〈k〉/gh
0
0.1
0.2 (f)
x∗ = 2.60m
(A)
x∗ = 0.00m
(B)
x∗ = 0.55m
(C)
x∗ = 1.00m
(D)
x∗ = 1.50m
(E)
x∗ = 2.00m
t/T0 0.2 0.4 0.6 0.8 1
(F )
x∗ = 2.60m
Figure 12: Phase-averaged k time series for the surf zone plunging breaking case TK2at (a − f) ∼ 4 cm and (A − F ) ∼ 9 cm above the bed at different cross-shore locationsafter the initial break point, x∗ = 0. Comparison between NHWAVE results with 4 σlevels (dashed lines), 8 σ levels (dotted-dashed lines), 16 σ levels (thick solid lines) andthe measurement (thin red solid lines)
56
Figure 13: Time-averaged normalized k field,√k/gh, for the surf zone spilling breaking
case TK1. NHWAVE results with (a) 4 σ levels, (b) 8 σ levels, and (c) 16 σ levels. Dashlines show the crest 〈η〉max, mean η and trough 〈η〉min elevations.
57
(z−η)/h
-1
-0.5
0
(a)
x∗ = -7.67m
(z−η)/h
-1
-0.5
0
(b)
x∗ = -0.46m
(z−η)/h
-1
-0.5
0
(c)
x∗ = 0.26m
√
k/gh
0 0.075 0.15
(z−η)/h
-1
-0.5
0
(d)
x∗ = 0.88m
(e)
x∗ = 1.48m
(f)
x∗ = 2.09m
(g)
x∗ = 2.71m
√
k/gh
0 0.075 0.15
(h)
x∗ = 3.32m
Figure 14: Time-averaged normalized k profiles for the surf zone spilling breaking caseTK1 at different cross-shore locations before and after the initial break point, x∗ = 0.Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines), 16 σ levels (solid lines) and the measurements (circle markers).
58
(z−η)/h
-1
-0.5
0
(a)
x∗ = -0.50m
(z−η)/h
-1
-0.5
0
(b)
x∗ = 0.00m
(z−η)/h
-1
-0.5
0
(c)
x∗ = 0.55m
√
k/gh
0 0.075 0.15
(z−η)/h
-1
-0.5
0
(d)
x∗ = 1.00m
(e)
x∗ = 1.50m
(f)
x∗ = 2.00m
√
k/gh
0 0.075 0.15
(g)
x∗ = 2.60m
Figure 15: Time-averaged normalized k profiles for the surf zone plunging breaking caseTK2 at different cross-shore locations before and after the initial break point, x∗ = 0.Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines), 16 σ levels (solid lines) and the measurements (circle markers).
59
d0 = 0.44m1 : 35
SWL
Wave Maker
x
z
Figure 16: Experimental layout of Bowen & Kirby (1994). Vertical solid lines: the cross-shore locations of the free surface measurements.
d0 = 0.47m1 : 20
SWL
Wave Maker
x
z
Figure 17: Experimental layout of Mase & Kirby (1992). Vertical solid lines: the cross-shore locations of the free surface measurements.
60
f/fp0.1 1 10
S(f)(cm
2s)
10-3
0.09
11
(c)
d/db =0.2
S(f)(cm
2s)
10-3
0.09
11
d/db =0.4
S(f)(cm
2s)
10-3
0.09
11
d/db =0.6
S(f)(cm
2s)
10-3
0.09
11
d/db =0.9
S(f)(cm
2s)
10-3
0.09
11
(b)
d/db =1.2
S(f)(cm
2s)
10-3
0.09
11
(a)
d/db =2.0
f/fp0.1 1 10
d/db =0.2
d/db =0.4
d/db =0.6
d/db =0.8
d/db =1.2
d/db =2.0
f/fp0.1 1 10
d/db =0.2
d/db =0.4
d/db =0.6
d/db =0.8
d/db =1.2
d/db =2.0
Figure 18: Power spectral density evolution, S(f) (cm2.s), for the random breaking cases,(a) BK with fp = 0.225Hz, (b) MK1 with fp = 0.6Hz, and (c) MK2 with fp = 1.0Hzat different cross-shore locations. Comparison between NHWAVE results with 4 σ levels(dashed lines), 8 σ levels (thick solid lines) and the corresponding measurements (circles).Here, d is the still water depth, and db is the still water depth at x = xb (db ∼ 20.5cmfor BK and db ∼ 12.5cm for MK1 and MK2). The solid lines show an f−2 frequencydependence.
61
100
300
500
(a1)
0
10
20 (a2)
1
3
5 (a3)
d (cm)0 10 20 30 40
-1.5
0
1.5 (a4)
100
300
500
(b1)
Nw
0
5
10 (b2)H1/10
Hm0
η
0.5
1.5
2.5 (b3)T1/10
Tm0
d (cm)0 10 20 30 40
-1.5
0
1.5 (b4)Skewness
Asymmetry
300
600
900(c1)
0
5
10 (c2)
0.5
1
1.5 (c3)
d (cm)0 10 20 30 40
-1.5
0
1.5 (c4)
Figure 19: Cross-shore variation of different Second- and third-order wave statistics for(a) BK, (b) MK1 and (c) MK2. Comparison between NHWAVE results with 4 σ levels(dashed lines), 8 σ levels (solid lines) and the corresponding measurements (circles). Here,Nw is the number of waves detected by the zero-up crossing method, H0.1 and T0.1 are theaveraged height and period of the one-tenth highest waves in the signal, Hm0
, Tm02are the
characteristic wave height and period based on the power spectra of the signal, Skewness=η3/(η2)3/2 > 0 is the normalized wave skewness, and Asymmetry= H(η)3/(η2)3/2 < 0 isthe normalized wave asymmetry. The results shown in (a) and (c) has the same label asin (b).
62
Figure 20: Time-averaged velocity field, u, for the surf zone irregular breaking case MK2.NHWAVE results with (a) 4 σ levels and (b) 8 σ levels. Dash lines show Hrms + η. Colorsshow u/
√gh.
Figure 21: Time-averaged normalized k field,√k/gh, for the surf zone irregular breaking
case MK2. NHWAVE results with (a) 4 σ levels and (b) 8 σ levels. Dash lines showHrms + η.
63
x(m)0 10 20 30 40 50 60 70 80 90
z(m
)
-4
-3
-2
-1
0
1 : 12
1 : 12
1 : 12−1 : 12
1 : 36
1 : 361 : 24
SWL
Wave Maker
Figure 22: Experimental layout of Scott et al. (2004). Vertical thick solid lines: the cross-shore locations of the velocity measurements. Vertical thin solid lines: the cross-shorelocations of the free surface measurements.
H(cm)
0
40
80 (a)
x(m)20 30 40 50 60 70 80
η(cm)
-5
0
5 (b)
Figure 23: (a) Cross-shore distribution of the wave height, H = 〈η〉max − 〈η〉min, and (b)mean water level, η, for the surf zone regular breaking waves on a barred beach case S1.Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines) and the measurements of Scott et al. (2004) (circle markers). Vertical lines:the cross-shore locations of the velocity measurements shown in Figure 22.
64
〈η〉−
η(m
)
-25
0
25
50 (a)
x = 32.70m
(e)
x = 58.31m
〈η〉−
η(m
)
-25
0
25
50 (b)
x = 40.01m
(f)
x = 61.97m
〈η〉−
η(m
)
-25
0
25
50 (c)
x = 51.00m
t/T0 0.2 0.4 0.6 0.8 1
(g)
x = 69.27m
t/T0 0.2 0.4 0.6 0.8 1
〈η〉−η(m
)
-25
0
25
50 (d)
x = 54.65m
Figure 24: Phase-averaged free surface elevations for the surf zone regular breaking waveson a barred beach case S1 at different cross-shore locations before and after the bar.Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines) and the measurement (thin red solid lines).
65
Figure 25: Time-averaged velocity field, u, for the surf zone regular breaking waves on abarred beach case S1. NHWAVE results with (a) 4 σ levels, and (b) 8 σ levels. Dash linesshow the crest 〈η〉max and trough 〈η〉min elevations. Colors show u/
√gh. Vertical lines:
the cross-shore locations of the velocity measurements shown in Figure 22.
66
(z−η)/h
-1
-0.5
0
(a)
x = 32.7m
(z−η)/h
-1
-0.5
0
(b)
x = 40.0m
(z−η)/h
-1
-0.5
0
(c)
x = 51.0m
u/√
gh-0.25 0 0.25
(z−η)/h
-1
-0.5
0
(d)
x = 54.6m
(e)
x = 58.3m
(f)
x = 62.0m
u/√
gh-0.25 0 0.25
(g)
x = 69.3m
Figure 26: Time-averaged normalized horizontal velocity (undertow) profiles for the surfzone regular breaking waves on a barred beach case S1 at different cross-shore locationsbefore and after the bar. Comparison between NHWAVE results with 4 σ levels (dashedlines), 8 σ levels (dotted-dashed lines), and the measurements at two different longshorelocations (open and solid circle markers). Red lines at (c) show the results 3m seaward ofthe corresponding measurement location.
67
Figure 27: Time-averaged normalized k field,√k/gh, for the surf zone regular breaking
waves on a barred beach case S1. NHWAVE results with (a) 4 σ levels, and (b) 8 σ levels.Dash lines show the crest 〈η〉max, mean η and trough 〈η〉min elevations. Vertical lines: thecross-shore locations of the velocity measurements shown in Figure 22.
68
(z−η)/h
-1
-0.5
0
(a)
x = 32.7m
(z−η)/h
-1
-0.5
0
(b)
x = 40.0m
(z−η)/h
-1
-0.5
0
(c)
x = 51.0m
√
k/gh
0 0.075 0.15
(z−η)/h
-1
-0.5
0
(d)
x = 54.6m
(e)
x = 58.3m
(f)
x = 62.0m
√
k/gh
0 0.075 0.15
(g)
x = 69.3m
Figure 28: Time-averaged normalized k profiles for the surf zone regular breakingwaves on a barred beach case S1 at different cross-shore locations before and after thebar.Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels(dotted-dashed lines), and the measurements (circle markers). Red lines at (c) show theresults 3m seaward of the corresponding measurement location.
69
f/fp0.1 1 10
S(f)(m
2s)
10-5
10-3
10-1
(d)
x = 54.6
S(f)(m
2s)
10-5
10-3
10-1
(c)
x = 51.0
f/fp0.1 1 10
10-5
10-3
10-1
(g)
x = 69.3
S(f)(m
2s)
10-5
10-3
10-1
(b)
x = 40.0 10-5
10-3
10-1
(f)
x = 62.0
S(f)(m
2s)
10-5
10-3
10-1
(a)
x = 32.7 10-5
10-3
10-1
(e)
x = 58.3
Figure 29: Power spectral density evolution, S(f) (m2.s), for the random breaking on abarred beach case S2 at different cross-shore locations. Comparison between NHWAVEresults with 4 σ levels (dashed lines), 8 σ levels (thick solid lines) and the correspondingmeasurements (circles). The solid lines show f−2.
70
η
-5
0
5 (a)
Hm0
0
20
40
60 (b)
Tm0
2
3
4 (c)
x (cm)20 30 40 50 60 70 80
Skew,Assy
-1.5
0
1.5 (d)
Figure 30: Cross-shore variation of different Second- and third-order wave statistics forthe random breaking on a barred beach case S2. Comparison between NHWAVE resultswith 4 σ levels (dashed lines), 8 σ levels (solid lines) and the corresponding measurements(circles). The definitions are the same as in Figure 19.
71
Figure 31: Time-averaged velocity field, u, for the random breaking on a barred beachcase S2. NHWAVE results with (a) 4 σ levels and (b) 8 σ levels. Dash lines show Hrms+η.Colors show u/
√gh.
72
(z−η)/h
-1
-0.5
0
(a)
x = 32.7m
(z−η)/h
-1
-0.5
0
(b)
x = 40.0m
(z−η)/h
-1
-0.5
0
(c)
x = 51.0m
u/√
gh-0.25 0 0.25
(z−η)/h
-1
-0.5
0
(d)
x = 54.6m
(e)
x = 58.3m
(f)
x = 62.0m
u/√
gh-0.25 0 0.25
(g)
x = 69.3m
Figure 32: Time-averaged normalized horizontal velocity (undertow) profiles for the ran-dom breaking on a barred beach case S2 at different cross-shore locations before and afterthe bar. Comparison between NHWAVE results with 4 σ levels (dashed lines), 8 σ levels(dotted-dashed lines), and the measurements (circle markers). Red lines at (c) show theresults 3m seaward of the corresponding measurement location.
73
Figure 33: Time-averaged normalized k field,√k/gh, for the random breaking on a barred
beach case S2. NHWAVE results with (a) 4 σ levels and (b) 8 σ levels. Dash lines showHrms + η.
74
(z−η)/h
-1
-0.5
0
(a)
x = 32.7m
(z−η)/h
-1
-0.5
0
(b)
x = 40.0m
(z−η)/h
-1
-0.5
0
(c)
x = 51.0m
√
k/gh
0 0.075 0.15
(z−η)/h
-1
-0.5
0
(d)
x = 54.6m
(e)
x = 58.3m
(f)
x = 62.0m
√
k/gh
0 0.075 0.15
(g)
x = 69.3m
Figure 34: Time-averaged normalized k profiles for the random breaking on a barred beachcase S2 at different cross-shore locations before and after the bar.Comparison betweenNHWAVE results with 4 σ levels (dashed lines), 8 σ levels (dotted-dashed lines), andthe measurements (circle markers). Red lines at (c) show the results 3m seaward of thecorresponding measurement location.
75
z∗
-0.075
0
0.075
t∗ =0.0
(a)
z∗
-0.075
0
0.075
t∗ =0.15
(b)
x∗
z∗
-0.075
0
0.075
t∗ =0.3
(c)
z∗
-0.075
0
0.075
t∗ =0.75
(d)
x∗
-0.5 -0.25 0 0.25 0.5 0.75 1
z∗
-0.075
0
0.075
t∗ =1.0
(e)
Figure 35: Snapshots of the free surface evolution during active breaking for the interme-diate depth breaking case, RM1. Comparison between NHWAVE results with 8 σ levels(thick solid lines) and the VOF-based model (thin solid lines). The free surface time seriesat the locations indicated by vertical dashed lines are shown in Figure 36.
t∗
-2 -1 0 1 2 3 4
η(m
)
-0.1
-0.05
0
0.05
0.1
0.15
x∗ =-0.31
(a)
t∗
-2 -1 0 1 2 3 4
x∗ =0.15
(b)
Figure 36: Time series of the free surface evolution for the intermediate depth breakingcase, RM1 at (a) before and (b) after the break point (x∗ = 0). Comparison betweenNHWAVE results with 8 σ levels (solid lines) and the corresponding measurements ofRapp & Melville (1990) (circles).
76
t (s)10 15 20 25 30 35
η(m
)
-0.05
0
0.05
x =11.1 m (f)
η(m
)
-0.05
0
0.05
x =9.4 m (e)
η(m
)
-0.05
0
0.05
x =7.2 m (d)
η(m
)
-0.05
0
0.05
x =6.0 m (c)
η(m
)
-0.05
0
0.05
x =4.0 m (b)
η(m
)
-0.05
0
0.05
x =1.8 m (a)
Figure 37: Time series of the free surface evolution at different x locations for the deepwater breaking case, T1. Comparison between NHWAVE results with 8 σ levels and thehorizontal resolution of ∆x = 10mm (dotted dashed lines) and the measurement of Tianet al. (2012) (solid lines).
77
x∗-3 -2 -1 0 1 2 3
(b)
x∗-3 -2 -1 0 1 2 3
η2/η2 1
0.5
0.6
0.7
0.8
0.9
1
(a)
Figure 38: Normalized time-integrated potential energy density, Ep, for the intermediatedepth breaking case, RM2. Comparison between the corresponding measurements (circles)and NHWAVE results with (a) 8 σ levels and (b) 16 σ levels, using different horizontalresolutions of ∆x = 23mm (solid lines), ∆x = 10mm (dashed lines) and ∆x = 5mm(dashed-dotted lines).
78
S(f)
10-6
10-3
100
x =1.84m (a)
S(f)
10-6
10-3
100
x =6.61m (b)
S(f)
10-6
10-3
100
x =7.23m (c)
S(f)
10-6
10-3
100
x =7.90m (d)
S(f)
10-6
10-3
100
x =9.41m (e)
f/fc0.1 1 10
S(f)
10-6
10-3
100
x =11.05m (f)
Figure 39: Energy density spectrum evolution, S(f) (cm2.s) for the deep water breakingcase, T1. Comparison between NHWAVE results with 8 σ levels using ∆x = 10mm (thicksolid lines) and ∆x = 5mm (dashed lines) as well as the measurements of Tian et al.(2012) (solid lines). Vertical dotted lines indicate the frequency range of the input packet.
79
〈u〉/C
-0.2
0
0.2 (a)
t∗-2 -1 0 1 2 3 4 5 6 7 8
〈w〉/C
-0.2
0
0.2 (b)
Figure 40: Normalized ensemble-averaged velocities for RM1 using 8 σ levels (dashed lines)and 16 σ levels (solid lines) at x∗ = 0.6, z∗ = −0.025. The circles are the measurementsof the corresponding case adopted from Rapp & Melville (1990), Figure 41.
u∗ lp
-0.05
0
0.05
z∗ =-0.016
(a) (b)
w∗
lp
-0.02
0
0.02
(c) (d)
u∗ lp
-0.05
0
0.05
z∗ =-0.031
w∗
lp
-0.02
0
0.02
u∗ lp
-0.05
0
0.05
z∗ =-0.078
w∗
lp
-0.02
0
0.02
t∗0 5 10 15
u∗ lp
-0.05
0
0.05
z∗ =-0.155
t∗0 5 10 15
t∗0 5 10 15
w∗
lp
-0.02
0
0.02
t∗0 5 10 15
Figure 41: Normalized low-pass filtered velocities for RM1 using 8 σ levels (dashed lines)and 16 σ levels (solid lines), at (a,c) x∗ = 0.15 and (b,d) x∗ = 0.60 at different elevations.The circles are the measurements of the corresponding case adopted from Rapp & Melville(1990), Figure 42.
80
x∗
0 0.5 1 1.5
z∗
-0.3
-0.15
0
(a)
u∗c
0 0.01-0.3
-0.15
0
(b)
M ∗
-0.001 0 -0.3
-0.15
0
(c)
x∗0 0.5 1 1.5
-0.3
-0.15
0
z∗
(d)
u∗c
0 0.01-0.3
-0.15
0
(e)
M ∗
-0.001 0 -0.3
-0.15
0
(f)
Figure 42: (a, d) Spatial distribution of the normalized mean current, u∗c; (b, e) normalized
horizontal-averaged mean current in the streamwise direction, u∗c and (c, f) normalized
accumulative horizontal-averaged mass flux, M∗, in the breaking region for RM1. (a-c)NHWAVE results with 8 σ levels and (d-f) LES/VOF results by Derakhti & Kirby (2014b).
81